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Merger, Ease of Entry, and Entry Deterrence in aDynamic Model
Anthony M. Marino and Ján Zábojník
USC Center for Law, Economics & Organization Research Paper No. C03-3
CENTER FOR LAW, ECONOMICSAND ORGANIZATION
RESEARCH PAPER SERIES
Sponsored by the John M. Olin Foundation
University of Southern California Law SchoolLos Angeles, CA 90089-0071
This paper can be downloaded without charge from the Social Science Research Networkelectronic library at http://ssrn.com/abstract=xxxxxx
Merger, Ease of Entry, and Entry Deterrence in a Dynamic Model∗
Anthony M. Marino
Marshall School of Business
University of Southern California
Los Angeles, CA 90089-1427
E-Mail: [email protected]
Ján Zábojník
Marshall School of Business
University of Southern California
Los Angeles, CA 90089-1427
E-mail: [email protected]
April 24, 2003
Abstract
This paper analyzes whether the ease and speed of entry of new firms can mitigate the
anti-competititve effects of a merger, in a dynamic model of endogenous merger. In our model,
if the new firms can enter quickly, it is more likely that merger is motivated by efficiency as
opposed to increased market power. Thus, there is less of a reason to challenge the merger. On
the other hand, if entry of new firms becomes less costly, firms may have a stronger incentive
to monopolize the industry through horizontal merger. We also show that when the incumbent
can engage in entry deterrence activities, anti-merger policy can decrease welfare.
Keywords: Horizontal Mergers, Entry Deterrence
JEL Classification: K21, L41
∗We would like to thank Tom Gilligan, Joe Harrington, John Matsusaka, Kevin J. Murphy, Eric Rasmusen,Mark Weinstein, as well as the seminar participants at Cornell University and the Summer CLEO Workshop at theUSC Law School, for helpful comments and suggestions. Both authors gratefully acknowledge the financial supportprovided by the Marshall General Research Fund.
1. Introduction
Suppose that two firms in an industry decide to merge. Under what conditions should they be
allowed to do so? This is a very old question, but neither the government regulators nor academic
economists have a clear-cut answer. The basic trade-off seems to be clear: The potential cost of
a merger is that it could increase the merged firms’ market power, which would allow them to
restrict output and increase prices. This would hurt consumers and create a deadweight loss. On
the other hand, the merged firms might be able to better utilize their combined resources, resulting
in more efficient production and potentially lower prices for consumers, if the efficiency gains are
large enough.
It turns out that the above trade-off is fairly hard to assess, both in practice and in theory. There
are many complicating factors — the efficiencies that the firms attribute to the proposed merger
may be possible to achieve through other means (say, joint venture), the merger may not result
in higher prices if the firms are able to implicitly collude even without merging, the merger may
invite new entry, and so on. In this paper, we concentrate on the effects of potential entry following
merger. The 1992 Horizontal Merger Guidelines issued jointly by the Federal Trade Commission
and the U.S. Department of Justice pay considerable attention to the effects of new entry. In fact,
a whole section (Section 3) out of a total of five sections is devoted to entry analysis. Moreover,
several recent lower court decisions make it clear that the courts take seriously the argument that
easy entry makes a proposed merger less harmful.1
The reason why the possibility of new entry is important is that if two firms merge, they may
1In United States v. Waste Management, Inc., 743 F.2d 976 (2nd Cir. 1984), the Second Circuit approved amerger between two Dallas commercial waste haulers that created a firm controlling 48.8% of the Dallas market (seePitofsky, 1990). The court based its decision on the claim that the Dallas market could be relatively easily entered,either by new entrants or from haulers established in nearby Fort Worth. For other decisions where ease of entryplayed an important role see, e.g., United States v. Syufy Enterprises, 903 F.2d 659 (9th Cir. 1990), United States v.Calmar, Inc., 612 F. Supp. 1298 (D.N.J. 1985), Laidlaw Acquisition Corp. v. Mayflower Group, Inc., 636 F. Supp.1513 (S.D. Ind. 1986), or Consolidated Gold Fields, PLC v. Anglo American Corp., 698 F. Supp. 487 (S.D.N.Y.1988).
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create room in the market for another firm, and the entry of a new firm may mitigate any anti-
competitive effects of the merger. This possibility is explicitly mentioned in the Guidelines: “In
markets where entry is ... easy ..., the merger raises no antitrust concern and ordinarily requires
no further analysis.” This policy is clearly fashioned so as to treat easy and speedy entry as a
mitigating factor in a merger challenge.
Our goal in this paper is twofold. First, we want to take the policy espoused in the Guidelines
at its face value and evaluate the argument that when new entry is easy and speedy, there is less
of a need to challenge a horizontal merger. Simple economic intuition would seem to support this
policy — easy and speedy entry should quickly eliminate the exertion of monopoly power of the
merged firms. However, we show that this intuition may be flawed. Second, we argue that in order
to fully assess the effects of an anti-merger policy, one has to take into account the alternative ways
in which firms may be able to acquire or protect their monopoly power. In particular, we study
the effectiveness of an anti-merger policy when firms can engage in entry deterrence.
We build and analyze a dynamic model of merger and entry. We allow for the possibility that
entry takes time, in order to capture the considerations expressed in the Guidelines, which recognize
that it is important to assess if the entry “can achieve significant market impact within a timely
period.” In particular, the firms in our model are infinitely lived and entry can occur only at the
beginning of each period. In such a dynamic model, the incumbent firms face a trade-off when
considering a merger with the purpose of gaining monopoly power: Until new entry occurs, they
can exercise their increased monopoly power and reap higher profits. However, the merger may
invite new firms into the industry, which makes monopolization short lived. Whether merger is
optimal or not then depends upon how quickly can new firms enter and how competitive is the
industry in the absence of mergers.
Our analysis of this trade-off yields some interesting results. First, we show that if the moti-
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vation behind the merger is solely to gain monopoly power (i.e., the merger does not generate any
efficiencies in production), then the incentive to merge decreases the faster are the new firms able to
enter the industry. In particular, if the new entrants can establish their operation sufficiently fast,
then mergers are never observed in this industry. Thus, the Guidelines’ focus on the quickness of
new entry is justified in our model, albeit for a slightly different reason than the one espoused in the
Guidelines. In our model, if the new firms can enter quickly, it is unlikely that a merger proposed
by the incumbents is motivated by the goal of increased market power. Rather, it probably has the
efficiency motivation and, therefore, there is less of a reason to challenge it.
Second, it turns out, rather surprisingly, that if there is a cost of entry that must be incurred by
new firms, then, for reasonable parameter values, a decrease in this cost can increase the incentive
of incumbent firms to gain monopoly power through merger. This is because low entry cost means
that the ‘free entry’ industry structure is relatively competitive, which makes the interim gain from
monopolization large. In such an industry, if the FTC follows the Guidelines and takes the ease of
entry as a mitigating factor, it is likely to approve mergers that have the purpose of monopolizing
the market and challenge those that are motivated by efficiency gains.
Finally, our analysis demonstrates that an important consideration in the assessment of an
anti-merger policy should be the ease with which the incumbents can prevent new entry through
deterrence activities. It might seem that this consideration is captured by the Guidelines in their
analysis of the entry effects, in particular in the discussion of ease of entry and entry costs. However,
the Guidelines appear to treat the entry costs as fixed and exogenously given. What we have in mind
here are endogenous entry barriers erected by the incumbents, such as advertisement expenditures,
investment in cost reducing technology, exclusive contracts with suppliers, or any other activities
identified in the rich literature on entry deterrence. Our point is that these endogenous barriers
to entry can depend upon the anti-merger policy. In particular, entry deterrence and merger are
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to some extent substitutes to firms that wish to preserve their monopoly power. Hence, if merger
becomes more costly or less likely, firms may optimally spend more resources on entry deterrence.
Because the resources spent on entry deterrence represent a waste, tough anti-merger policy may
decrease overall welfare. We use this argument to demonstrate that if entry deterrence is feasible
and costly, ease of entry in our model always represents a mitigating factor in merger. The reason
is that if the industry is to be monopolized, it is less wasteful to monopolize it through merger
rather than through entry deterrence.
The literature on the analysis of horizontal mergers is vast with most models characterized by
a fixed number of firms, nonendogenous merger, and a static time frame. The key papers in this
category are the contributions by Salant, Switzer and Reynolds (1983), Deneckere and Davidson
(1985), Perry and Porter (1985), and Farrell and Shapiro (1990). These papers are generally
concerned with the effects of merger on price and surplus levels and the appropriate public policy
toward mergers. Saloner (1987) shows that pre-merger predatory pricing by the incumbent can
favorably affect the incumbent’s terms of takeover and decrease the possibility of entry. In a model
due to Reitzes and Levy (1995), the ability to price discriminate can lead to welfare reducing
mergers, which do not promote entry. Rasmusen (1988) considers a static and exogenous model of
merger through the buy-out of entrants and shows that the prospect of subsequent buy-out may
encourage otherwise unprofitable and inefficient entry. Kamien and Zang (1990), (1991) and (1993)
consider endogenous models of merger and discuss the limits to monopolization through acquisition,
but they too retain the assumptions of a fixed time frame and a fixed number of firms. Horn and
Persson (2001) use notions of coalition formation from cooperative game theory to formulate an
endogenous model of the merger process. However, this analysis, like the work of Kamien and
Zang, employs a static model.
The papers more directly related to our work are those by Werden and Froeb (1998), Cabral
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(1999) and Spector (2001), who all study the question of whether the industry equilibrium price
increases or decreases as a result of a merger, when there exists a possibility of subsequent new
entry. In addition, in a model of product differentiation and price competition, Cabral (1999)
derives the interesting result that consumer welfare may be lower if the cost efficiencies created by
merger are greater.
While each of the above three models studies entry subsequent to merger, each is a one period
model focused on the effects of exogenous merger on consumer welfare. However, as was persuasively
argued by Gowrisankaran (1999), policy makers need dynamic models with endogenous mergers
“to correctly answer even simple antitrust questions.” Gowrisankaran offers such a model and
studies the occurrence of and the effects of mergers on endogenously determined entry, exit, and
investment. The model is sufficiently complex that it cannot be analytically solved and requires
the use of numerical methods.2
Our model, like Gowrisankaran’s ambitious model, is a dynamic endogenous model of merger.
We are able to solve the model analytically, because we have kept it simple by focusing our analysis
on mergers motivated purely by the desire to monopolize the industry (that is, our mergers generate
no efficiency gains) and by adopting simplifying assumptions regarding the merger process. On the
other hand, we enrich the model by allowing the incumbent firm to spend resources on entry
deterrence. Our analytic solution allows us to study the comparative dynamics of the speed of
entry and the cost of entry on the incentive to merge. Further, we are able to analyze the effects
of anti-merger policy when entry deterrence activities are not completely prohibited.
The rest of the paper proceeds as follows. In the next section we describe the model and discuss
our main assumptions. Section 3 provides the analysis of the case where entry deterrence is not
possible. Section 4 allows entry deterrence and explores its effects. Section 5 concludes. All proofs
2The first dynamic models of merger were presented by Berry and Pakes (1993) and Cheong and Judd (1992).They, too, use numeric methods to analyze their models.
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are provided in the Appendix.
2. The Model
The industry. Consider an infinitely lived industry that is currently served by one incumbent
firm, I.3 Time is continuous, but new firms can only enter at the beginning of discrete time periods,
t, of length M. At the beginning of each period t, the incumbent I faces a threat of (instantaneous)
entry from many potential entrants, who decide sequentially4 whether to enter in that period or
not. If a firm i decides to enter at the beginning of period t, it has to incur a one-time entry cost
F > 0. We will adopt the tie breaking rule that a firm enters the industry only if the expected
profit after entry is strictly greater than the cost of entry, F . All firms in this economy discount
the future using a common interest rate, r.
Profits. Let M be the instantaneous monopoly profit (i.e., the monopoly profit per unit of time)
in this industry, so that if no entry ever occurs, then I’s present value of all future profits is
M(∞) = R∞0 Me−rτdτ = M/r. If at a given point in time exactly k firms operate in this industry,
then each of them earns an instantaneous profit of πk (thus, π1 = M). We will assume that πk
decreases in k, i.e., πk+1 < πk, for each k. In addition, the combined industry profit is at most
equal to the monopoly profit: M ≥ kπk for any k.5
Note that as in Salant, Switzer and Reynolds (1983), merger in this framework is equivalent to
shutting down all but one of the merging firms. In other words, all firms in our model are always
3Although most of our qualitative results (including Propositions 1, 3, and 4) would also obtain in a two-periodmodel, such a model would not necessarily be more tractable.
4 It is well known that if entry is simultaenous, the model can exhibit both pure and mixed strategy equilibria.The only purpose of the assumption that entry is sequential is to eliminate the mixed strategy equilibria.
5We are deliberately avoiding tying our analysis to a specific model of oligopolistic competition, like Cournot orBertrand models. We would like to note though, that both of the above assumptions, πk+1 < πk and M > kπk forall k, are satisfied in a wide range of models. For example, it can easily be checked that the first property is metin the traditional Cournot model with linear demand and cost. The property that M > kπk for any k is also metin the general Cournot model, under strictly concave profit functions for each firm and the assumption that eachfirm’s output is a strategic substitute for any other firm’s output. See, for example, Tirole (1989), pp. 218-219.
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identical with respect to their profitability, independent of whether they were created through
a merger or established as new entrants. This means that there are no efficiency benefits to
mergers in our model, so that any merger must be motivated purely by the prospect of increased
market power (in general, this requires that firms have constant marginal costs and their products
are homogeneous). While we are aware that many real world mergers are driven by efficiency
considerations, our goal here is to evaluate the argument that ease of entry should be a mitigating
factor in challenging a given merger. In order to focus on this argument, we find it useful to abstract
from other possible mitigating factors, like efficiency gains. Nevertheless, we are confident that the
model could be extended to allow for many forms of efficiency gains from merger, without affecting
our qualitative results.
Entry deterrence. At the beginning of each period t, that period’s incumbent(s) can engage in
deterrence activities to discourage entry in period t. Since entry deterrence considerations do not
affect our analysis until Section 4, we defer the description of the entry deterrence process to that
section.
Merger proposals. At the beginning of any period t in which the industry contains k ≥ 2
firms (including I), the firms can negotiate a merger (as long as mergers are not prohibited by the
government). The merger bargaining has at most two rounds in any given period. The bargaining
starts by I making a merger offer to all firms that it decides to buy out in that period.6 It is
assumed that if the offer is rejected by any of the target firms (or if I chooses not to make any
merger offer), the first round of bargaining is unsuccessful and all k firms have to wait for a length
of time τ < M before one of them can make a new offer.7 During that time, each of them receives
6We will assume that whenever I merges with some other firm or firms, the resulting firm is firm I, even if I didnot make the merger offer (although such a merger never happens in equilibrium). This assumption is inconsequential— we only adopt it to be specific.
7Thus, the merger offers are contingent on the acceptance by all targeted firms. By allowing contingent offers, weare following Stigler’s (1950) suggestion for resolving the free-rider problem in the merger bargaining problem he was
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a stream of πk profits. After the period of time τ , the second round of bargaining starts by Nature
choosing randomly one of the k firms to make a new contingent offer. This offer, too, has to be
accepted by all targeted firms; if any of them rejects, the bargaining process is over for the rest of
the period.8 At the beginning of period t+1, if the industry contains at least two firms (including
new entrants, if any), the bargaining process starts anew, with I making the first offer.
Solution concept. LetKt be the set of all firms (including new entrants) present in the industry at
the beginning of period t. If a firm is chosen to make a merger proposal, its strategy is characterized
by the set of firms, J ⊂ Kt, to which it extends the buy-out offers, and by the buy-out prices,
Pi ≥ 0, it offers each firm i ∈ J . (In our analysis below, we do not use the buy-out prices explicitly.
Rather, we couch our discussion in terms of the shares of the merger surplus captured by merger
participants.) The strategy set of potential entrants in any given period consists of only two actions
— enter and do not enter. The solution concept we employ is that of Markov Perfect Equilibrium
(MPE), where the state variable is the number of firms, k, that are present in the industry at a
given time.
3. The merger decision when entry deterrence is not possible
Before we proceed with the analysis, it is useful to derive the expressions for the firms’ one pe-
riod profits, as well as for their (potential) total lifetime profits. Analogous to the case of a
permanent monopoly, the present value of total profits of each of k permanent oligopolists is
first to identify. This free rider problem is caused by the fact that, if there are relatively many firms in an industry,some of them may have an incentive to reject a merger offer hoping to benefit more if other firms merge and increasethe industry’s concentration and price. See Kamien and Zang (1990) and Gowrisankaran and Holmes (2000), amongothers, for more recent investigations of this free rider problem. As recognized by Stigler, this problem vanishes iffirms can make contingent merger offers.
8Alternatively, we could have assumed that the bargaining process continues until an offer is accepted by alltargeted firms, rather than ending after two rounds of bargaining. This would not change our results, although itwould complicate the analysis because it would be necessary to consider how the bargaining procedes if it is notconcluded by the end of the period (an out of equilibrium event) and if new firms enter at the beginning of the nextperiod, t+ 1.
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πk(∞) =R∞0 πke
−rτdτ = πk/r. Similarly, the one period profit of each of k oligopolists is
πk(M) =R M0 πke
−rτdτ = πkr [1−e−rM] = πk
r [1−δ(M)], where δ(M) ≡ e−rM is the one period cumulative
discount factor.9 Thus, πk(∞) = πk(M)1−δ(M) .
In this section, we analyze the firms’ entry and merger incentives under the assumption that
entry deterrence is not possible. We start with the following simple example, in which the intuition
can be seen most easily.
3.1. An example
Suppose that the industry can accommodate at most 2 firms. Suppose also, for illustration purposes
in this example, that I has all the bargaining power vis a vis the other firms (as will become clear
later, this corresponds to a τ close to M). In a one period setting (i.e., the industry only exists
for a single period of length M), the buy out decision is very simple: The entrant sells out at the
beginning of the period if and only if it is offered at least π2(M) and I buys the entrant out whenever
M(M)−π2(M) > π2(M), or π2(M) < M(M)2 . Because this inequality is frequently satisfied in standard
oligopoly models, in a one-period framework, we typically expect a monopoly-preserving merger.
In a dynamic setting, the situation is more complicated. If the entrant does not sell out, then it
expects to get π2(M) in every period, so that the total present value of its profits is π2(∞) = π2(M
)/[1− δ(M)]. This is the minimum it will accept when I proposes a merger. If I wants to preserve
its monopoly position, it has to buy out a new entrant at the beginning of every period. Merger
will therefore be observed if and only if M(M)1−δ(M) − π2(M)
[1−δ(M)]2 >π2(M)1−δ(M) , or
M(M)− 2− δ(M)1− δ(M)π2(M) > 0.
The expression on the left hand side is the total surplus created by the merger in any given period,
9Again, we will use M(M) instead of π1(M).
9
and the condition says that the industry will be monopolized if and only if this surplus is positive.
Clearly, when δ(M) is relatively small (the interest rate is high or the period ∆ is long), then the
surplus is high and the above condition is easily satisfied. On the other hand, if the interest rates
are low or if I faces a frequent threat of entry, then it is better off accommodating the entrant,
rather than trying to preserve its monopoly position through merger, period after period.
3.2. The general case
Suppose now, more generally, that after any new entry, the number of firms present in this industry
at the beginning of a period t, but before the start of merger negotiations, is kt ≥ 2. For lack of
a better term, we will call kt the "pre-merger number of firms". The equilibrium in this model
is characterized by (i) the equilibrium pre-merger number of firms in every period, k∗, (ii) the
post-merger number of firms, (iii) the time of merger (if any) in every period (that is, whether the
merger occurs at the beginning of the period or after a delay of the length τ), and (iv) the shares of
the merger surplus captured by the firms participating in the merger. Let V I(k∗) and V E(k∗) be
the equilibrium present values of total profits of the incumbent I and of an entrant E respectively.
In equilibrium, it must be that the pre-merger number of firms, k∗, is such that, given the insider
firms’ merger strategies and other firms’ entry strategies, entry is profitable for k∗ firms, but not
for k∗+1 firms, i.e., V E(k∗+1) ≤ F < V E(k∗). In addition, each firm’s merger strategy maximizes
this firm’s present value of profits, conditional on the new firms’ entry strategies and other firms’
merger strategies. All the strategies are stationary in the sense that they can only depend upon
the number of firms in the industry, but not on the firms’ behavior in the past periods.
Thus, suppose that I buys out j ≤ kt − 1 competitors in period t. (While the relevant proofs
allow for the out of equilibrium possibility that kt − j > k∗, in the following discussion we will
assume that kt − j ≤ k∗, to streamline the exposition.) Then the sum of the present values of all
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j + 1 firms that merged in period t is given by
πkt−j(M) + δ(M)V I(k∗).
That is, if the j + 1 firms merge into a single firm, I, this firm becomes one of kt − j firms for the
rest of the period and then, at the beginning of the period t+ 1, has the present value of V I(k∗).
On the other hand, if there is no merger, the total value of the j + 1 firms is
(j + 1)πkt(M) + δ(M)[V I(k∗) + jV E(k∗)].
In this case, each firm competes in period t as one of k oligopolists and then, at the beginning of
the next period, gets its equilibrium continuation value. (Recall that the firms not merging are, at
the beginning of period t+ 1, equivalent in terms of profitability to new entrants.) The difference
between the above two expressions represents the surplus, Sjkt(M), created by period-t merger:
Sjkt(M) = πkt−j(M)− (j + 1)πkt(M)− δ(M)jV E(k∗).
We now solve the period-t bargaining game through backward induction and characterize (in
lemmas 1 and 2 below) the firms’ optimal merger strategies conditional on the pre-merger number
of firms, kt. Suppose that the first round of bargaining failed and, after the period of time τ , Nature
chose a firm i to make the final merger offer in this period. At this point, the remaining surplus
from a merger involving j + 1 firms is given by10
Sjkt(M −τ) = πkt−j(M −τ)− (j + 1)πkt(M −τ)− δ(M −τ)jV E(k∗). (1)
10 It can be checked that this expression captures the merger surplus even if the merger does not include firm I.
11
Note that if Sjkt(M −τ) > 0 it decreases in τ and that Sj
kt(M −τ) → Sj
kt(M) as τ → 0. This
corresponds to a situation in which I has no special advantage over the other firms in merger
bargaining. On the other hand, Sjkt(M −τ) → −δ(M −τ)jV E(k∗) < 0 for all j as τ →M. This
means that if the bargaining delay, τ , is too long (longer than some τ∗ <M), any benefits from
merger get dissipated in the event of a disagreement in the first round. In this case, I has all the
bargaining power.
Lemma 1. Suppose that kt ≤ k∗ and Sjkt(M −τ) ≥ 0 for a j < kt − 1. Then SM
kt(M −τ) ≡
Skt−1kt
(M −τ) > Sjkt(M −τ).
Lemma 1 says that, if positive, the second round merger surplus is maximized when the industry
is monopolized. Note that this result also holds for the first round surplus, Sjkt(M), because the
lemma allows for τ = 0.
Since the second round of bargaining is final, firm i that was chosen by Nature to make the
merger proposals in this round in effect makes a take-it-or-leave-it offer to its targets. Hence, it
captures the whole surplus, Sjkt(M −τ). Because, by Lemma 1, this surplus is maximized when all
firms merge, firm i monopolizes the whole market whenever SMkt(M −τ) ≥ 0, otherwise, it makes
no merger offer. Now, every firm in the industry is equally likely to be chosen by Nature to make
the second round offer. Hence, if the first round of bargaining fails, then in the second round each
firm gets, in expectation, max{0, SMkt(M−τ)kt
}. This discussion leads to the following result.
Lemma 2. Suppose that kt ≤ k∗.
(a) If SMkt(M) > 0, the industry will be monopolized in period t, otherwise, no merger will occur
in that period.
(b) Suppose that SMkt(M) > 0, i.e., the industry is monopolized in period t. If τ ≤ τ∗, each firm
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i 6= I captures the share βkt =δ(τ)kt
SMkt(M−τ)
SMkt(M) of the merger surplus SM
kt(M). The incumbent, I,
captures the share αkt = 1− (kt − 1)βkt . If τ > τ∗, then αkt = 1 and βkt = 0.
(c) If SMk∗ (M) > 0, then SM
k∗ (M) =1−δ(M)
1−δ(M)αk∗hM − k∗−δ(M)
1−δ(M) πk∗(M)i.
According to Lemma 2, if in a given period the pre-merger number of firms is not higher than
the equilibrium number k∗, then any merger observed in this period will be a merger to monopoly.
Moreover, the industry will be monopolized if and only if the merger surplus is positive. Note that
this conclusion does not depend upon the firms’ relative bargaining powers. Because the merger
bargaining process is efficient, the firms reach an agreement whenever merger creates a surplus.
We now proceed by investigating the decision problems of potential entrants. For this, we first
need to introduce some new notation. When deciding whether to enter, the firms anticipate the
outcome of the merger process, conditional on the pre-merger number of firms, kt. Define V E1 (kt)
as the expected present value of profits (gross of the cost of entry, F ) earned by a period-t entrant,
given that there is merger to monopoly in period t, the pre-merger number of firms in that period
is kt, and ks = k∗ for all s ≥ t+ 1. Let V E2 (kt) be defined similarly, but assuming that there is no
merger in period t, and let V E(kt) ≡ max{V E1 (kt), V
E2 (kt)}. Thus, V E(kt) is the value of entry in
period t if the pre-merger number of firms ends up to be kt in that period and the firms’ merger
strategies are as described in Lemma 2. Next, let k0 be the free entry number of firms in this
industry if no mergers are possible, i.e., πk0+1(M) ≤ F [1− δ(M)] < πk0(M). Analogously, define k
by V E(k + 1) ≤ F [1− δ(M)] < V E(k). Finally, let nt−1 denote the (post-merger) number of firms
that is in the industry at the end of period t − 1. Using these definitions, the following lemma
characterizes the optimal entry strategies of new firms.
Lemma 3. (a) If nt−1 < k, then exactly k − nt−1 new firms enter in period t. Otherwise, no
new firms enter in that period.
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(b) In every period, the pre-merger equilibrium number of firms is k∗ = k ≥ k0.
Lemma 3 is just a version of the standard result regarding the free entry number of firms in an
industry: firms keep entering as long as the expected present value of total profits after entry can
cover the fixed cost of entry, F . The lemma also says that the equilibrium pre-merger number of
firms is at least as high as when mergers are not feasible. This is because if merger is observed in
equilibrium, it must weakly increase each firm’s lifetime profit.
Given the entry decisions of firms, described in Lemma 3, and given the firms’ merger behavior,
described in Lemma 2, we are now in a position to identify the conditions under which merger
will be observed in this industry and characterize the equilibrium market structure. In particular,
Lemma 2 implies that the industry will be monopolized in equilibrium if and only if the equilibrium
merger surplus SMk∗ (M) is positive, i.e.,
M(M)− k∗ − δ(M)1− δ(M) πk∗(M) > 0. (2)
The following proposition recasts condition (2) in terms of the model’s more primitive parameters,
namely, in terms of the free entry number of firms in the absence of mergers, k0.
Proposition 1. Let D(k) ≡M− k−δ(M)1−δ(M)πk(M), let k0 be as defined in Lemma 3, and let SM
k0+1(M)
be as given in Lemma 2(c).
(i) In the unique equilibrium of this game, the incumbent either buys out all entrants in every
period or accommodates all of them in the first period, after which there is no new entry and
no merger in subsequent periods.
(ii) If D(k0) > 0, the industry is monopolized in every period. If D(k0) ≤ 0, the industry is
monopolized in every period if and only if πk0+1(M) + βk0+1SMk0+1(M) > F [1− δ(M)].
14
The entry strategies of new firms described in Lemma 3 mean that if I wants to preserve its
monopoly position, it has to buy out all of the entrants in every period. This attracts k∗ − 1
new firms to enter in the subsequent period and so on. Part (i) in Proposition 1 shows that the
incumbent either struggles every period to preserve her monopoly position or gives up completely.
It is never optimal in our framework for I to buy out only some entrants. This result is driven
by the fact that the total industry profit is higher under monopoly than under any other industry
structure.
Remark 1. Because Proposition 1 assumes that there are no governmental restrictions re-
garding merger, the number of real world firms pursuing buy-out strategies consistent with this
proposition must necessarily be limited. However, one can find supporting evidence from the late
1800’s, before antitrust policy was instituted. For example, in the late 1800’s, the practice of mo-
nopolization through acquisition of entrants was adopted by Standard Oil Company and American
Tobacco.11 More recent examples include Microsoft and Intel, who during the 1990’s through the
present (with less than rigorous enforcement of antitrust) have both pursued a vigorous strategy of
buying out new entrants of importance.12
Remark 2. While, empirically, merger to monopoly does not appear to be the prevalent case,
some of the most important merger cases where the ease of entry played a major role involved
companies that post-merger unambiguously dominated their respective markets. For example, in
United States v. Calmar, Inc. the district court, using the ease of entry argument, declined to
enjoin the merger of Calmar and Realex, two producers of plastic sprayers, whose post-merger
market share in the regular sprayer market would have been 83%, with HHI more than 7,100. In
11See Standard Oil Co. of New Jersey v. United States, 221 U.S. 1 (1911) and McGee (1958). See also UnitedStates v. American Tobacco Co., 221 U.S. 106 (1911) and Burns (1986).12See www.microsoft.com/msft/invest.htm for a listing of Microsoft’s acquistions since 1994. Also see
www.netaction.org/msoft/world/table/html for a table showing more specifics regarding these buyouts. Fora description of Intel’s acquisitons, see the company’s acquisitions and capital portfolio web page athttp://www.intel.com/intel/finance/acquisitions/.
15
Laidlaw Acquisition Corp. v. Mayflower Group, Inc. the FTC declined to challenge Laidlaw’s
attempted takeover of Mayflower. In the Pacific Northwest, the takeover would have increased
Laidlaw’s share of the market for private contract bus services to school children to 85.9%. Again,
the FTC’s decision was based primarily on the ease of entry argument. As our final example, we
mention the United States v. Syufy Enterprises case. In 1981, Syufy entered the first-run movie
market in Las Vegas and through a series of acquisitions came to dominate 90% of the market
(reduced to 75% by the time of the suit). Using the ease of entry argument, the Ninth Circuit
concluded that there was no violation of section 7.13
Condition (2) preceding Proposition 1 shows that in our dynamic setting the incumbent is less
likely to use merger to preserve her monopoly position than in a static, one period setting. (In a one
period setting, merger is optimal whenever M(M) ≥ k∗πk∗(M), which is always true in our model.)
The reason is that the side effect of a merger is that it invites new entry in the future. Thus, the
merger can only guarantee a monopoly position for one period, until new entrants appear, but the
buy out price for each entrant reflects the entrant’s potential lifetime profit.
3.3. The effects of the speed of entry
The first implication of Proposition 1 is that the incentives to monopolize an industry through
merger depend upon how quickly can new firms establish their operation in this industry. This
relationship is shown formally in the following proposition.
Proposition 2. There exists a M∗∈ (0,∞) such that the industry is monopolized in every period
if and only if M>M∗.
According to Proposition 2, the industry is monopolized through merger only if new firms can-
not enter fast enough (that is, if M>M∗). Otherwise, the incumbent is better off being one of k∗
13See, e.g., Pitofsky (1990) for further details.
16
oligopolists, rather than buying out new entrants every period. As mentioned in the Introduction,
this result lends some support to the recommendation found in the 1992 Merger Guidelines, ac-
cording to which the FTC should assess if entry “can achieve significant market impact within a
timely period.” However, the reason given in the Guidelines is just the initial part of our argument.
According to the Guidelines, timely entry mitigates the anti-competitive effects of the merger. In
our analysis, the reasoning does not stop there: Because the anti-competitive effects of merger are
quickly mitigated, the firms do not have an incentive to merge in the first place. Hence, if a merger
is initiated in such an industry, it is most likely because it generates production efficiencies (which
we do not model here) — it cannot be motivated by market power. However, it is worth noting that
this does not necessarily mean that such a merger should always be approved, because, regardless
of motivation, it does create monopoly power.
3.4. The effects of the cost of entry
Throughout the rest of the paper we will keep the time period length M fixed. Thus, in order to
economize on notation, from now on we will drop the argument M and write M instead of M(M),
πk instead of πk(M), and so on.
The second implication of Proposition 1 is that, as condition (2) shows, the incentives to merge
for the purpose of creating a monopoly depend upon the resulting industry structure if the merger
does not occur. This is reflected through profit πk∗ that would be earned by each of the firms in
every period in the absence of a merger. According to Lemma 3, this profit is in turn determined
by the ease of entry, as captured by the fixed cost of entry F. At a first blush, one might think
that the lower is the cost of entry, the lower is the equilibrium industry concentration that we
should expect to prevail in this economy in the absence of any anti-merger policy. However, as the
following proposition demonstrates, this intuition is incorrect in the dynamic setting of the present
17
model.
Proposition 3. Suppose that each firm’s profit declines sufficiently fast with the number of firms
in the industry, i.e., (k − δ)πk decreases in k ≥ 2. Then as the cost of entry, F , decreases,
the industry is more likely to be monopolized through merger in every period.
According to Proposition 3, if each individual firm’s profit decreases sufficiently fast with the
number of firms in the industry, and if mergers are allowed, then the industry is more likely to
be monopolized if the cost of entry is relatively low. Note that the condition of Proposition 3 is
implied by the stronger condition that the total profit of entrants, (k − 1)πk, is decreasing in k,
and it is equivalent to this condition if δ is close to 1. The intuition for Proposition 3 is most
easily seen for the case where I has all the bargaining power, i.e., βk∗ = 0: If the cost of entry is
low then, in the absence of merger, the industry would be fairly competitive. This has two effects
on I’s incentives to merge. First, if the profit per firm decreases fast with the number of firms,
the incumbent’s alternative to merger (namely, being one of k∗ oligopolists) is fairly unattractive.
Second, the fast decline in profits per firm makes it relatively cheap to buy out all entrants, because
the total buy-out price is (k∗−1)πk∗1−δ and the condition of the proposition slows any rate of increase
in this buy-out price with respect to increases in k∗.
The importance of the result in Proposition 3 can be assessed by contrasting it with the 1992
Merger Guidelines. According to the Guidelines, ease of entry should be a mitigating factor in the
challenge of a given merger.14 The implication seems to be that the lower is the sunk cost of entry,
the less the need to prohibit merger. Our result shows that by following this guideline, the FTC
might be led to approve mergers that are, in spite of the fact that new entry is easy, motivated purely
14The notion of ‘ease of entry’ as used in the Guidelines is somewhat vague, as it is primarily defined in terms ofthe effects the new entry has on the degree of industry competitiveness. However, one of the steps the Guidelinespropose for gauging the effects of the new entry is the assessment of the profitability of new entry in the presence ofsunk entry costs. The ease of entry in our model is measured by sunk entry costs.
18
by the intent to monopolize the industry (without any gains in efficiency). Similarly, if entry is
relatively difficult, in our model this may not be enough to make a merger profitable, unless there
are substantial efficiency gains. Hence, challenging a merger proposal in such an industry, just
because entry is relatively difficult, would be misguided.
Example.
Clearly, the relevance of Proposition 3 depends on how likely it is that the condition on profits
holds. To demonstrate that this condition can hold in non-trivial environments, we now show that
it is satisfied for the case where the firms compete as Cournot oligopolists with linear demand and
cost. Let market demand per period be given by p = a− b(P
qi), where qi denotes the output of
firm i in a given period. If each firm has cost sqi, then per firm profit πk is given by πk =h
(k+1)2,
where h ≡ (a−s)2b . Using these expressions, the condition (k − δ + 1)πk+1 < (k − δ)πk becomes
k2+ k(1− 2δ) ≥ (1+ 3δ). It can easily be checked that with any δ ∈ [0, 1], the condition is met for
all k ≥ 3. The condition is satisfied for k = 2, if δ < 5/7.
To illustrate the proposition, assume again that βk∗ = 0 and note that in this case it must
always be that k∗ = k0, because the entrants do not expect to share in any potential merger
surplus. Note also that, in this example, condition (2) holds if and only if
(1− δ)k∗2 − 2(1 + δ)k∗ + 3δ + 1 > 0. (3)
If we suppose that the cost of entry is some F1 ∈ [ h16(1−δ) ,
h9(1−δ)), then the equilibrium pre-merger
number of firms is k∗ = 2 and (3) holds if and only if δ < 1/5. Next suppose that the cost of
entry decreases to some F2 ∈ [ h25(1−δ) ,
h16(1−δ)). In this case k
∗ = 3, so that (3) holds if and only if
δ < 1/3. Thus, for discount factors δ ∈ (15 , 13), the industry is not monopolized when the cost of
entry is F1, but it becomes monopolized when the cost of entry decreases to F2.
19
We now add entry deterrence as an alternative to merger and turn our attention to I’s choice
of the method for preserving her monopoly position.
4. Merger or entry deterrence?
In reality, merger is not the only tool that firms can use to preserve or gain monopoly power. If
merger is likely to be challenged, then the incumbents may instead focus on driving their existing
rivals out of the industry or preventing new firms from entering. Since the logic is the same in
both of these cases, we will couch our analysis in terms of entry deterrence. We show here that the
optimal anti-merger policy should take into account the possibility that the incumbents will step
up their entry deterrence activities if they believe merger is not a feasible alternative, in which case
a tough anti-merger policy can decrease overall welfare.
There are many possible activities a monopolist can engage in to try to prevent entry. This
may include writing exclusive contracts with input suppliers, brand proliferation, limit pricing,
vertical integration with a supplier, and so on. We will not model the specific nature of these
activities — enough has been written about these by other authors. We will simply assume that these
activities are costly to the incumbents and can represent a waste of resources. Examples of wasteful
entry-deterrence activities include holding extra production capacity or investing excessively in cost
reducing technology and R&D. We formalize entry deterrence in a simple way: We assume that
in any given period, the incumbents can completely deter entry by spending the total amount of
K > 0 on deterrence activities and that any amount smaller than K will not discourage a potential
entrant.15 If at the beginning of a given period there are n ≥ 1 incumbents, they share the amount
K equally, i.e., each has to pay K/n. Formally, this would be equivalent to a technology that
allows the incumbents to increase the new firms’ cost of entry, F , to a level at which no new entry
15This formalization of entry deterrence is similar to that used, for example, in Bernheim (1984).
20
is profitable, i.e., k∗ = 1. If entry is not deterred in a given period, the industry can still be
monopolized in that period, through merger.
The timing of events that occur at the beginning of each period is summarized as follows:
1. The incumbents decide whether to spend K on deterrence activities.
2. If K is spent, then no entry occurs in that period. If less than K is spent, then new firms
can enter as long as they find the entry profitable.
3. If mergers are not prohibited, industry insiders enter merger negotiations.
The next lemma demonstrates that our analysis of the merger behavior derived in the previous
section continues to hold in the current framework.
Lemma 4. Suppose that entry is not deterred in equilibrium. Then the incumbent either buys out
all entrants in every period or accommodates all of them in the first period, after which there
is no new entry and no merger in subsequent periods.
In order to be able to assess the welfare effects of I’s trade-off between merger and entry
deterrence, we first need to introduce the social cost of entry deterrence. When I spends an amount
K on entry deterrence activities, this amount represents I’s dissipated profit, but not all of that is
a deadweight loss. Some entry deterrence activities, such as limit pricing, result in consumer gains.
To capture this situation, we define the social cost of entry deterrence as the difference K −∆CS,
where the term ∆CS ≥ 0 represents a gain in consumer surplus induced through entry deterrence.
The case ∆CS = 0 represents a situation where the deterrence activity is completely wasteful.
Due to the stationary nature of the game, if entry deterrence is an optimal strategy in some
period, then it is optimal in every period. Hence, we need to consider two cases — one in which I
deters entry in every period and one in which I never deters entry.
Case 1: I does not deter entry
21
Since this is the same problem as the one analyzed in the previous section, Proposition 1
immediately applies. That is, I buys out all entrants or otherwise it buys out none of them. In
this case, I’s total discounted profit if entry is not deterred and merger is optimal is given by
V IND(k
∗) =πk∗ + αk∗S
Mk∗
1− δ,
where the subscript ND stands for “no deterrence”. That is, in every period, I gets the oligopoly
profit πk∗ , plus its share of the merger surplus, SMk∗ .
Case 2: I deters entry
If I decides to deter entry, its total profits are given by
V IED =
M −K
1− δ.
Here, the subscript ED stands for “entry deterrence”.
First, entry deterrence can only be profitable if V IED ≥ πk∗
1−δ , that is, K < M − πk∗ , because
otherwise I would be better off allowing k∗ − 1 firms to enter and accommodating them. Second,
comparing V IND(k
∗) with V IED we find that monopolization through entry deterrence is more prof-
itable than monopolization through merger ifK ≤ K1 ≡M−πk∗−αk∗SMk∗ =
(1−αk∗)M−(1−αk∗k∗)πk∗1−δαk∗ .
Using this preliminary analysis, we obtain the following result.
Proposition 4. Let K2 ≡ (k∗− 1)F +∆CS. Suppose that (2) holds and entry deterrence cannot
be prevented by the government.
(a) If K1 < K < K2, then antitrust laws that prohibit mergers increase social welfare.
(b) If Kmax ≡ max{K1,K2} < M − πk∗ and K ∈ (Kmax,M − πk∗), then antitrust laws that
prohibit mergers decrease social welfare.
22
(c) For all other K, antitrust laws that prohibit mergers have no effect on social welfare.
While sometimes anti-merger laws can be socially beneficial, part (b) in Proposition 4 shows
that, if entry deterrence cannot be prevented, prohibiting mergers can decrease social welfare. When
entry deterrence cost is relatively high (above the threshold Kmax), entry deterrence becomes an
inefficient substitute for merger.16
To relate the results of Proposition 4 to our previous analysis and to the role of the ease of
entry in anti-merger policy, fix some K 0 and assume that it is smaller than M − πk∗ , so that entry
deterrence can be profitable. We are interested in expressing part (b) in the proposition in terms
of the ease of entry, F . This part applies when K1 and K2 are both smaller than K 0. It is clear
that K2 = (k∗ − 1)F +∆CS is small when F is small. Now suppose that the incumbent has all
the bargaining power, i.e., αk∗ = 1. Then K1 =(k∗−1)πk∗
1−δ , which tends to be small for small F if
the entrants’ combined profits, (k∗ − 1)πk∗ , decline fast with the number of firms. On the other
hand, if the entrants’ combined profits do not decrease fast enough with k∗ or if the incumbent’s
bargaining power is relatively small, then small F implies small K2 but large K1.17 In this case,
Proposition 3 says that anti-merger policy has no effect.
Hence, when wasteful entry deterrence is not prohibitively costly and the cost of entry is rela-
tively low, antitrust laws that preclude mergers tend to decrease social welfare (or have no effect)
in our model. This logic is very different from that in the Guidelines. The Guidelines consider easy
entry as a mitigating factor in merger because it is presumed that the resulting monopoly power
16A related question is how successful are the anti-merger laws in creating a relatively competitive environment inthe economy. The implicit assumption behind the FTC Guidelines seems to be that the application of anti-mergerlaws always results in a less concentrated industry structure. Is this assumption justified? Due to space considerations,we do not address this question in the present article. However, in an earlier version of this paper we demonstratethat for a range of parameter values, anti-merger policy can lead to a more concentrated industry than a laissez-fairepolicy.17To see this, suppose that I’s bargaining power is the smallest possible, i.e., the same as the bargaining power of
entrants (recall that this case corresponds to τ = 0). Then K1 ≡ M − πk∗ − βk∗SMk∗ = M − (1 − δ)V E
1 (k∗), which
increases in k∗, because, as was shown in the proof of Proposition 1, V E1 (.) is a decreasing function. Therefore, in
this case K1 is large when F is small.
23
would be partially offset due to likely subsequent entry. As we have shown in Proposition 3, this is
argument is incomplete, because when the costs of entry are low, firms inside the industry may have
a stronger incentive to buy out new entrants than when the entry costs are high. However, when
firms can engage in entry deterrence activities, it is optimal in our framework to permit mergers if
the cost of entry is low, in order to avoid a waste of resources spent by the incumbents on entry
deterrence activities.
The policy implications of our model can then be summarized as follows. If it is hard to prevent
the incumbents from engaging in entry deterrence activities (or in activities aimed at driving existing
firms out of the industry), easy entry should always be considered as a mitigating factor in merger.
However, if entry deterrence is not very wasteful or can easily be prevented, then Proposition 3
applies and low entry costs are not an unambiguous mitigating factor. In particular, the ease of
entry defence should be discounted if the profit per firm decreases fast with the number of firms in
the industry.
5. Conclusion
In this paper we take a closer look at the effects of entry on firms’ incentives to monopolize an
industry through horizontal merger. The importance of entry of new firms (or expansion by ex-
isting rivals) in assessing the effects of a horizontal merger has been recognized both in academic
literature (e.g., Werden and Froeb, 1998, Cabral, 1999, Spector, 2000) and in the 1992 Merger
Guidelines issued jointly by the U.S. Justice Department and FTC. We show that the Guidelines’
recommendation that if new entry is relatively fast there is less need to challenge a merger, makes
sense in our dynamic model, although our reasoning is somewhat different from the one found in
the Guidelines. The Guidelines suggest that if the new entry is fast, it can quickly mitigate any
anti-competitive effect of the merger. We point out that if this is true, then the incumbents cannot
24
gain by merging unless there are substantial synergies created by the merger. Hence, if a merger is
initiated in an industry where new entry would follow relatively quickly, then it is probably because
the merger would enhance the merged firms’ efficiency — if it did not, it would not be attempted in
the first place. This point is similar to the one made independently by Spector (2001). However,
Spector’s argument is based on a one period model and focuses on entry costs rather than the time
dimension of entry (speed of entry).18
Our second result is less supportive of the conclusions reached in the Guidelines. We demon-
strate that a decrease in exogenous entry costs can increase the motivation of the incumbent firms
to monopolize the market through horizontal merger. The alternative would be to operate in a
highly competitive industry implied by low entry costs, which makes merger look more attrac-
tive. In such an industry, following the Guidelines can lead the FTC to approve mergers aimed at
monopolizing the market and challenge those that generate efficiency gains.
The last point that we make is that a merger challenge decision should not be made without
considering the incumbents’ alternative means of preserving or gaining market power. Merger may
well be the most efficient of all these alternatives, and prohibiting it without preventing the firm
from channelling its monopolization efforts elsewhere could be counter-productive.
18Spector’s result is a generalization of the idea found in Werden and Froeb (1998).
25
Appendix: Proofs
Proof of Lemma 1: We have SMkt(M −τ) =M(M −τ)− ktπkt(M −τ)− δ(M −τ)(kt − 1)V E(k∗) >
πkt−j(M −τ)−(j+1)πkt(M −τ)−δ(M −τ)jV E(k∗) = Sjkt(M −τ) if (kt−j−1)πkt−j(M −τ)−(kt−j−
1)πkt(M −τ)−δ(M −τ)(kt−j−1)V E(k∗) > 0, where we have usedM(M −τ) ≥ (kt−j)πkt−j(M −τ).
Since kt − j − 1 > 0, the inequality in the lemma holds because πkt−j(M −τ) − πkt(M −τ) − δ(M
−τ)V E(k∗) > Sjkt(M −τ) ≥ 0. ¥
Proof of Lemma 2: Parts (a) and (b). If SMkt(M) < 0, then, by Lemma 1, Sj
kt(M) < 0 for all j.
Therefore, no mergers are initiated, because at least one participating firm would have to be made
worse off if the merger goes through.
Thus, suppose that SMkt(M) > 0. Assume first that τ < τ∗. Then also SM
kt(M −τ) > 0, and,
by the argument preceding the lemma, if the bargaining fails in the first round, the firm that gets
to make the merger offers in the second round captures the whole remaining surplus, SMkt(M −τ).
Because every firm has the same chance of making the second round offer, and because the second
round comes only after the period of time τ , the expected present value of getting into the second
round is δ(τ)SMkt
(M−τ)kt
for every firm. Thus, I has to offer at least this amount to every firm it
decides to buy out (being a profit maximizer, I offers exactly this amount). Therefore, I’s value
if it buys out j competitors is πkt(M) +·Sjkt(M)− jδ(τ)
SMkt(M−τ)kt
¸+ δ(M)V I(k∗), where the term in
the brackets represents the part of the merger surplus captured by I. This term is maximized at
j = kt − 1, that is, when the industry is monopolized, because
SMkt (M)− (kt − 1)δ(τ)
SMkt(M −τ)kt
> Sjkt(M)− jδ(τ)
SMkt(M −τ)kt
(*)
for any j < kt−1. To see this, rewrite the inequality as SMkt(M)−Sj
kt(M) > (kt− j−1)δ(τ)S
Mkt(M−τ)kt
,
26
which holds if
(j + 1)SMkt (M)− ktS
jkt(M) > 0,
because δ(τ) ≤ 1 and SMkt(M −τ) ≤ SM
kt(M). Substituting for SM
kt(M) and Sj
kt(M), we obtain
(j + 1)SMkt (M)− ktS
jkt(M) = (j + 1)[M(M)− ktπkt(M)− δ(M)(kt − 1)V E(k∗)]
−kt[πkt−j(M)− (j + 1)πkt(M)− δ(M)jV E(k∗)]
= (j + 1)M(M)− ktπkt−j(M)− (kt − j − 1)δ(M)V E(k∗)
≥ (kt − j − 1)[jπkt−j(M)− δ(M)V E(k∗)] > (kt − j − 1)Sjkt(M).
Hence, (*) holds if Sjkt(M) ≥ 0, because j < kt − 1. If Sj
kt(M) < 0, then the right hand side of (*) is
negative, while the left hand side is positive (due to SMkt(M) > 0), so that (*) again holds.
Thus, in this case, I’s profit maximizing strategy in the first round of bargaining is to offer
δ(τ)SMkt
(M−τ)kt
to every firm in the industry. Given that no firm can do better by rejecting the offer,
each firm accepts in the first round. This proves the claim that if SMkt(M) > 0 and τ < τ∗, the
industry is monopolized, each firm i 6= I gets δ(τ)SMkt
(M−τ)kt
= δ(τ)kt
SMkt(M−τ)
SMkt(M) SM
kt(M) = βktS
Mkt(M)
in addition to its disagreement value, and the incumbent receives SMkt(M) − δ(τ)
(kt−1)SMkt (M−τ)kt
=
[1− (kt − 1)βkt ]SMkt(M) = αktS
Mkt(M).
Finally, suppose that SMkt(M) > 0 and τ ≥ τ∗. Then SM
kt(M −τ) < 0, which means that if the
bargaining fails in the first round, it is not worth resuming it in the second round. Hence, every
firm approached by I in the first round accepts any buyout price higher or equal to its disagreement
value, πkt(M)+δ(M)V E(k∗). A profit-maximizing incumbent will therefore offer exactly this amount
in the first round and each firm will accept. Thus, I captures the whole merger surplus Sjkt(M).
Because, Sjkt(M) is maximized at j = kt − 1 (Lemma 1), I again buys out all of its competitors.
Part (c). Suppose that SMk∗ (M) > 0, i.e., merger is optimal if the equilibrium pre-merger number
27
of firms is k∗. Then in every period, k∗ − 1 new firms enter, to be subsequently bought out by
I. Each entrant’s discounted present value is V E(k∗), while I’s value is V I(k∗). Because the total
value of discounted profits in this industry, if monopolized, is M1−δ(M) , it must be that
M
1− δ(M) = V I(k∗) + (k∗ − 1)V E(k∗) + δ(M)(k∗ − 1)V E(k∗) + δ(M)2(k∗ − 1)V E(k∗) + ...
= V I(k∗) +(k∗ − 1)1− δ(M)V
E(k∗).
Substituting for V I(k∗) = πk∗(M)+αk∗SMk∗(M)1−δ(M) and V E(k∗) = πk∗ (M)+βk∗SMk∗ (M)
1−δ(M) , and using αk∗+ (k∗ −
1)βk∗ = 1, we obtain SMk∗ (M) =
1−δ(M)1−δ(M)αk∗
hM − k∗−δ(M)
1−δ(M) πk∗(M)i. ¥
Proof of Lemma 3: (a) If kt ≤ k∗ and SMkt(M) =M(M)−ktπkt(M)−δ(M)(kt−1)V E(k∗) > 0, there
is merger to monopoly (Lemma 2) and the entrants share the resulting surplus, SMkt(M), according
to their bargaining powers, βkt . In this case, the value after entry is VE1 (kt), given by
V E1 (kt) = πkt(M) + δ(M)V E(k∗) + βktS
Mkt (M).
When SMkt(M) ≤ 0, no merger is expected and the value after entry is V E
2 (kt), given by
V E2 (kt) = πkt(M) + δ(M)V E(k∗).
If kt > k∗, there are three possibilities. Let Sjkt(M) denote the merger surplus if kt > k∗ and I buys
out j firms in period t, and let βkt be the share of this surplus that goes to a firm i 6= I involved
in the merger. The first possibility is that no merger creates a positive surplus, i.e., Sjkt(M) ≤ 0 for
all j. In this case, the value of entry is V E3 (kt), given by
V E3 (kt) =
πkt(M)1− δ(M) ,
28
because if merger is not profitable in period t, it is not profitable in any subsequent period. More-
over, no new firms enter in subsequent periods, because the number of firms in the industry is
already greater than the equilibrium number, k∗ (and k∗ is unique, as will be shown in part (b)).
Now, due to the possibility that the entrant captures a share of the merger surplus (if positive), it
must be that V E(k∗) ≥ πk∗(M)1−δ(M) . Hence, V
E2 (kt) ≥ πkt(M) + δ(M) πk∗(M)1−δ(M) . For kt > k∗, this implies
V E2 (kt) > πkt(M) + δ(M) πkt(M)1−δ(M) = V E
3 (kt). Hence, VE3 (kt) < V E
2 (kt) < F for kt > k∗.
The second possibility is that Sjkt(M) is maximized and positive for some j ≥ kt−k∗. In this case,
the same proof as in Lemma 1 shows that SMkt(M) > Sj
kt(M), which means that the value of entry is
V E1 (kt). The final possibility is that S
jkt(M) is maximized and positive for some j∗ < kt−k∗. In this
case, no new firms enter in subsequent periods, because the number of firms remaining after merger
is already greater than the equilibrium number, k∗. Also, no new mergers will occur, because if
another merger could create positive surplus, then Sjkt(M) could not have been maximized at j∗.
The value of entry in this case is therefore given by
V E4 (kt) = πkt(M) + βktS
j∗kt(M) + δ(M)πkt−j
∗(M)1− δ(M) ,
where Sj∗kt(M) = πkt−j∗(M) − (j∗ + 1)πkt(M). That is, in this case the benefit of the time-t merger
is given by the extra profits the j∗ + 1 firms earn in period t as a result of the merger — if they
did not merge in period t they would earn πkt(M) each and then merge in period t+ 1. But πkt(M
)+βktSj∗kt(M) = βktπkt−j∗(M)+[1−(j∗+1)βkt ]πkt(M) < βktπkt−j∗(M)+(1−βkt)πkt(M) ≤ πkt−j∗(M),
which means that V E4 (kt) <
πkt−j∗(M)1−δ(M) = V E
3 (kt − j∗).
Now, let k0t be the largest kt such that V E1 (kt) > F and k00t the largest kt such that V E
2 (kt) > F
and let kmaxt = max{k0t, k00t }. Backward induction implies that the pre-merger number of firms
29
in period t will be at least kmaxt .19 At the same time, by construction of kmaxt , it must be that
V E1 (kt) < F and V E
2 (kt) < F for all kt > kmaxt . Moreover, V E3 (kt) < F and V E
4 (kt) < F because
for kt > k∗, V E3 (kt) < V E
2 (kt) < F, as shown above, and V E4 (kt) < V E
3 (kt − j∗) < F, where the
last inequality follows from kt − j∗ > k∗. Thus, consider a kt > kmaxt . If kt ≤ k∗, then the value
of entry is either V E1 (kt) or V
E2 (kt) which are both less than F , for kt < kmaxt . If, on the other
hand, kt > k∗, then the value of entry is either V E3 (kt) or V
E4 (kt), which were shown to be less
than F, so entry is again not profitable. Therefore, once the industry contains at least kmaxt firms,
no additional firms will enter in period t. Moreover, exactly kmaxt − nt−1 firms enter if nt−1 <
kmaxt . Finally, by construction, kmaxt is given by V E(kmaxt + 1) ≤ F [1− δ(M)] < V E(kmaxt ), where
V E(kt) = max{V E1 (kt), V
E2 (kt)}. This means that kmaxt = k (which is time independent, because
the function V E(.) is the same in every period).
(b) By the arguments above, in equilibrium it must be kt = kmaxt = k in every period. Hence,
k∗ = k in every period. Now, using kt = k∗ for all t, we get V E1 (k
∗) = πk∗(M)+βk∗SMk∗(M)1−δ(M) and
V E2 (k
∗) = πk∗(M)1−δ(M) . Hence, V
E(k∗) ≥ V E2 (k
∗) for all possible k∗. Because k∗ = k, k∗ is given by
V E(k∗ + 1) ≤ F [1− δ(M)] < V E(k∗), while k0 is defined by V E2 (k
0 + 1) ≤ F [1− δ(M)] < V E2 (k
0).
From V E(k∗) ≥ V E2 (k
∗), for all k∗, and because both V E(k∗) and V E2 (k
∗) are decreasing in k∗, it
must be that k∗ ≥ k0. ¥
Proof of Proposition 1: Start with part (ii). If no merger occurs in equilibrium, then the
equilibrium pre-merger number of firms is k∗ = k0 and the value of entry is
V E2 (k
0) =πk0(M)1− δ(M) .
19That is, suppose that the industry contains kmaxt −1 firms. Then at least one more firm has an incentive to enter,because by the definition of kmaxt , the value of entry for that firm is V E(kmaxt ) ≡ max{V E
1 (kmaxt ), V E
2 (kmaxt )} > 0 .
Next suppose that the industry contains kmaxt − 2 firms. Then the entry value of firm kmaxt − 1 is also V E(kmaxt ) > 0,because it expects that if it enters then, subsequently, firm kmaxt will enter, too. Moving backwards in this fashion,one can show that if the industry contains n < kmaxt firms, exactly kmaxt −n firms have an incentive to enter, becausefor each of them, the anticipated value of entry is V E(kmaxt ) > 0.
30
On the other hand, if there is merger in equilibrium, then the value of entry is
V E1 (k
∗) =πk∗(M) + βk∗S
Mk∗ (M)
1− δ(M) =πk∗(M) + δ(τ)
k∗ SMk∗ (M −τ)
1− δ(M) .
We will first show that V E1 (k
∗) decreases in k∗. To see this, substitute for SMk∗ (M −τ) from (1) to
obtain
V E1 (k
∗)[1− δ(M)] = πk∗(M) +δ(τ)
k∗[M(M −τ)− k∗πk∗(M −τ)− δ(M −τ)(k∗ − 1)V E
1 (k∗)],
where we have used the fact that in this case V E(k∗) = V E1 (k
∗). Rearranging and using πk∗(M) =
πk∗r [1− δ(M)] and δ(τ)δ(M −τ) = δ(M), we have
V E1 (k
∗)[1− δ(M)k∗
] =δ(τ)
k∗M(M −τ) + πk∗(τ).
Since the right hand side decreases in k∗ and 1 − δ(M)k∗ increases in k∗, it must be that V E
1 (.) is a
decreasing function of k∗.
Next, notice that k∗ > k0 if and only if V E1 (k
0+1) > F . To see this, note that if V E1 (k
0+1) ≤ F ,
then V E1 (k) ≤ F for all k ≥ k0 + 1 because V E
1 (.) is a decreasing function. This means that
V E(k) ≤ F for all k > k0, where V E(k) is the value function defined in Lemma 3. Therefore, it
cannot be that k∗ > k0. On the other hand, if V E1 (k
0 + 1) > F , then V E(k0 + 1) > F , which
implies that k∗ > k0.
Now, let k∗ = k0, and note that V E1 (k
0) > V E2 (k
0) if and only if SMk0 (M) > 0, that is, if and
only if D(k0) > 0, because according to Lemma 2(c), the equilibrium merger surplus is given by
SMk∗ (M) =
1−δ(M)1−δ(M)αk∗
hM − k∗−δ(M)
1−δ(M) πk∗(M)i= 1−δ(M)
1−δ(M)αk∗D(k∗). Suppose first that D(k0) > 0 and
assume that there is no merger in equilibrium. Because D(k0) > 0, it must be that SMk0 (M) > 0,
31
which means that the firms have an incentive to merge if the equilibrium number of them enter.
This contradicts the assumption that there is no merger in equilibrium; hence, if D(k0) > 0, the
equilibrium includes merger.
Next assume that D(k0) ≤ 0. First, suppose that πk0+1(M) + βk0+1SMk0+1(M) > F [1 − δ(M)].
Then V E1 (k
0 + 1) > F, which, as shown above, means that k∗ > k0. This in turn implies that
there must be merger in equilibrium, because otherwise it would have to be that V E(k∗) = V E2 (k
∗),
which would imply k∗ = k0. Finally, suppose that πk0+1(M) + βk0+1SMk0+1(M) ≤ F [1− δ(M)]. This
means that V E1 (k
0 + 1) ≤ F , which, from above, implies that k∗ = k0. But D(k0) ≤ 0 means that
also SMk0 (M) = SM
k∗ (M) ≤ 0, so that there is no merger in equilibrium.
Part (i). The part of this claim regarding I’s equilibrium buy-out strategy follows directly from
part (a) in Lemma 2. To see that the equilibrium is unique (up to the identity of entrants), note
that kmax in the proof of Lemma 3 is unique by construction. Because k∗ = kmax, the equilibrium
number of firms, k∗, is also unique. ¥
Proof of Proposition 2: The industry is monopolized if and only if condition (2) preceding
Proposition 1 holds. We will first show that, holding all other parameters fixed, an increase in M
makes this condition more likely to hold, i.e., the left hand side of (2) (LHS(2)) weakly increases
in M. Using πk/r = πk(M)1−δ(M) , condition (2) can be rewritten as
πk∗ [k∗ − δ(M)]− M [1− δ(M)] < 0. (2’)
Note also that k0 is independent of M, because πk0+1(M) ≤ F [1− δ(M)] < πk0(M), which defines k0,
is equivalent to πk0+1 ≤ Fr < πk0 . We want to show that the LHS(2’) weakly decreases in M.
Suppose first that an increase in M does not change k∗. Then differentiating the LHS(2’) with
respect to M yields ∂LHS(2’)∂M = δ0(M)[M − πk∗ ] < 0 for all M, where the inequality follows because
32
δ0(M) < 0. Next assume that k∗ changes with M. Then there are two possibilities. First, k∗ was
equal to k0 before M increased. Since it must be that k∗ ≥ k0 and k0 does not change with M, k∗
must have increased, so that now k∗ > k0. But this can only hold if the industry is monopolized
(see the proof of Proposition 1). Hence, in this case (2’) must be satisfied after an increase in M,
whereas it may have or may not have been satisfied before. Since this argument holds for any M,
this implies that, in this case, LHS(2’) weakly decreases in M.
The second possibility is that k∗ > k0 before the increase in M. In this case, it must be that
V E(k∗) = V E1 (k
∗) = πk∗(M)+βk∗SMk∗(M)1−δ(M) > F . But the function V E
1 (k∗) is continuous in M. Hence the
inequality V E1 (k
∗) > F is preserved if M increases slightly. But this means that k∗ does not change
with a slight increase in M, which is the case we already considered. This concludes the proof of
the claim that LHS(2’) weakly decreases in M.
Now, setting M= 0 yields δ(M) = 1, so that LHS(2’) = πk∗ [k∗ − δ(M)] > 0 for any k∗, which
means that (2’) cannot hold. On the other hand, when M→∞, then δ(M)→ 0 so that LHS(2’)→³πk∗k
∗ − M´< 0, in which case (2’) is satisfied for all k∗. Hence, there must exist a M∗∈ (0,∞)
such that (2’) holds if and only if M>M∗. ¥
Proof of Proposition 3: To prove the claim, it is enough to show that (2) is more likely to hold
as F decreases. First, note that if (k− δ)πk decreases in k then M − k∗−δ1−δ πk∗ < M − k∗+1−δ
1−δ πk∗+1.
That is, LHS(2) increases in k∗. Therefore, the proposition holds if k∗ increases as F decreases.
But this relationship holds, because both V E1 (k
∗) = πk∗+βk∗SMk∗1−δ and V E
2 (k∗) = πk∗
1−δ are (weakly)
decreasing functions of k∗. For V E2 (k
∗), this claim is obvious; for V E1 (k
∗) it was proved in the proof
of Proposition 1. Hence, the entry value function V E(.) = max{V E1 (.), V
E2 (.)} (defined in Lemma
3) is a (weakly) decreasing function, which means that k∗ (weakly) decreases in F . ¥
Proof of Lemma 4: Consider a period t in which I starts out as a monopoly. Suppose that in
this period exactly k∗ − n firms merge, where 0 < n < k∗. Then at the beginning of period t + 1
33
the industry starts with n ≥ 2 incumbents. Because we know from Lemma 2 and Proposition 1
that, in the absence of entry deterrence, I either buys out all entrants or there is no merger, the
only reason why, in the current setting, I would accommodate the n − 1 entrants is that these
firms would share some of the future entry deterrence costs. That is, if entry deterrence in the
future is never profitable, we are back to the scenario of the previous section, where Proposition 1
applies. On the other hand, if entry deterrence is optimal, then the profit to I when it is one of n
incumbents must be bigger than when I is the sole incumbent, because accommodating the n− 1
firms at time t is costly (according to Proposition 1, I’s period-t profit would be higher if either all
or none of the entrants were bought out).
Now, with n incumbents sharing K equally in period t + 1, that period’s profit to each of
them when they deter entry is πn − Kn (because there is no subsequent merger). But πn − K
n <
nπn−K < M−K, where the last inequality follows from nπn ≤M. Thus, whenever entry deterrence
is profitable, I would be strictly better off as a sole incumbent at time t + 1, deterring entry to
defend its monopoly position, rather than sharing the costs of entry deterrence in preserving an
oligopoly with n ≥ 2 firms. It therefore cannot be that, in equilibrium, fewer than k∗ firms merge
in a given period and the claim follows. ¥
Proof of Proposition 4: First, note that if the monopoly is preserved through entry deterrence,
the deadweight loss in that period is equal to K −∆CS (in addition to the deadweight loss from
monopoly, which does not depend upon the method of monopoly preservation). If I uses merger,
then the deadweight loss is equal to that period’s total costs of entry, (k∗ − 1)F. Hence, entry
deterrence is more efficient than merger if and only if K ≤ K2 ≡ (k∗ − 1)F +∆CS. We thus have
the following result: Suppose (2) holds, K < M − πk∗ , and there is no government regulation.
(i) If K ≤ min{K1,K2}, then I efficiently uses entry deterrence to preserve monopoly. (ii) If
K1 < K < K2, then I uses merger to preserve monopoly, even though it would be more efficient
34
to use entry deterrence. (iii) If K2 < K < K1, then I uses entry deterrence to preserve monopoly,
even though it would be more efficient to use merger. (iv) If K ≥ Kmax ≡ max{K1,K2}, then I
efficiently uses merger to preserve monopoly.
Part (a) in the proposition follows from (ii) above, because prohibiting mergers under these
parameter values induces I to switch from using merger to using entry deterrence, which increases
efficiency. Part (b) follows because if mergers are not feasible, I uses entry deterrence whenever
K < M − πk∗ . The result is then obtained immediately from (iv) above, because for K > Kmax
it would be more efficient to use merger to preserve monopoly, rather than using entry deterrence.
For part (c), note that the rest of the parameter values fall either under (i) or (iii) above. In
these cases, I prefers using entry deterrence to merger, so prohibiting merger has no effect on its
behavior. ¥
35
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